RV T-stats and home ones as well often get their power (if electronic) from the FURNACE (Blue wires on an RV T-stat normally) not the A/C.
Both home and RV rigs work like that
HOWEVER
Some RV types use replacable dry cell batteries.
And the one I have. needs no power it's an old fashion bi-metal spring with contascts (Furnace only)
Home was where I park it. but alas the.
2005 Damon Intruder 377 Alas declared a total loss
after a semi "nicked" it. Still have the radios
Kenwood TS-2000, ICOM ID-5100, ID-51A+2, ID-880 REF030C most times